Approximately transitive dynamical systems and simple spectrum

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Approximately transitive dynamical systems and simple spectrum

El Houcein El Abdalaoui, Mariusz Lemanczyk

To cite this version:

El Houcein El Abdalaoui, Mariusz Lemanczyk. Approximately transitive dynamical systems and

simple spectrum. 2009. �hal-00470162�

APPROXIMATELY TRANSITIVE DYNAMICAL SYSTEMS AND SIMPLE SPECTRUM

E. H. EL ABDALAOUI AND M. LEMA ´ NCZYK

Abstract. For some countable discrete torsion Abelian groups we give ex- amples of their finite measure-preserving actions which have simple spectrum and no approximate transitivity property.

AMS Subject Classifications (2000): 37A25

Key words and phrases: ergodic theory, dynamical system, AT property, funny rank one, Haar spectrum

1

Research supported by the EU Program Transfer of Knowledge “Operator Theory Methods

for Differential Equations” TODEQ and the Polish MNiSzW grant N N201 384834

2 E. H. EL ABDALAOUI AND M. LEMA ´ NCZYK

1. Introduction

The property of approximate transitivity (AT) has been introduced to ergodic theory by Connes and Woods [5] in 1985 in connection with their measure-theoretic characterization of ITPFI hyperfinite factors in the theory of von Neumann alge- bras. Since then the approximately transitive dynamical systems have been objects of study in several papers, e.g. [1], [9], [12], [13], [15], [16], [21]. All systems with the AT property have zero entropy [5], [8] and it is only recently [1], [9] that explicit examples of zero entropy systems without AT property have been discovered.

In view of the form of its definition (see below) the AT property seems to be related to spectral properties, or more precisely to the spectral multiplicity of the measure-preserving system. Indeed, it was already in the 1990. when J.-P. Thou- venot using some generic type arguments observed that the AT property implies the existence of a cyclic vector in the L ¹ -space of the underlying dynamical system.

Moreover, a certain modification of the definition of the AT property (in which we replace L ¹ -norms by L ² -norms) which was done in [16] implies L ² -simplicity of the spectrum. However, in general it is still an interesting open problem whether the original AT property implies simplicity of L ² -spectrum, i.e. simplicity of the spec- trum of the classical Koopman representation of the dynamical system. Moreover, as noticed for example in [9], neither the other implication was known to hold. The aim of this note is to answer this latter question – we give examples of systems with are not AT but have simple L ² -spectrum.

One more motivation for this note is the problem of a measure-theoretic charac- terization of simplicity of L ² -spectrum. This problem has a long history in ergodic theory, see for example [19], [20] and the references therein and our examples bring the answer to one more open question (stated explicitly in case of Z-actions in [11]

as well as in [9]): namely we give the negative answer to the question whether the class of funny rank one systems [10], [18] coincides with the class of systems with simple L ² -spectrum; indeed a funny rank one system enjoys the AT property (see below).

We would like however to emphasize that in the note we do not consider Z-actions which are the most popular objects of study in ergodic theory, and therefore the problems we mentioned above stay open for the class of Z-actions (and actions of many other groups). The systems which are considered below are actions of some torsion Abelian countable discrete groups. More precisely G = L _∞

n=0 Z/p n Z where (p n ) n ≥ 0 is an increasing sequence of primes numbers. We will then deal with so called Morse extensions given by some special Z/2Z-valued cocycles over a discrete spectrum action of G studied by M. Guenais in [14] (in that paper she proved that the resulting G-actions have Haar component of multiplicity one in the L ² -spectrum). All such systems have simple spectrum. We will show that for each sufficiently fast choice of parameters in Guenais’ construction we can apply the criterion of being non-AT system formulated in [1] (this criterion is an elabo- rated version of the method already used in [9]). More precisely, we will prove the following.

Theorem 1.1. Assume that G = L _∞

n=0 Z/p n Z with (p n ) n ≥ 0 an increasing sequence of primes numbers and p n ≥ 5 ²⁽ⁿ⁺¹⁾ , n ≥ 0. Then there exists a simple L ² -spectrum action of G without the AT property.

Following Connes and Woods [5] we now recall the definition of the AT property.

Let G be an Abelian countable discrete group. Assume that this group acts as measure preserving maps: g 7→ T g ∈ Aut(X, B , µ), where (X, B , µ) is a stan- dard probability Borel space. The action (X, B , µ, T ) with T = (T g ) g ∈ G (or simply T ) is called AT if for an arbitrary family of nonnegative functions f 1 , . . . , f l ∈ L ¹ ₊ (X, B , µ), l ≥ 2, and any ε > 0, there exist a positive integer s, elements g 1 , . . . , g s ∈ G, real numbers λ j,k > 0, j = 1, . . . , l, k = 1, . . . , s and f ∈ L ¹ ₊ (X, B , µ) such that

(1) k f j −

X s k=1

λ j,k f ◦ T g

k 1 < ε, 1 ≤ j ≤ l.

Recall now that T has the funny rank one property [10], [18] if for every A ∈ B and ε > 0 we can find F ∈ B , g 1 , . . . , g N ∈ G such that the family R = { T g

F, . . . , T g

F } , called a funny Rokhlin tower, consists of sets which are disjoint and for some J ⊂ { 1, . . . , N }

(2) µ A △ [

i ∈ J

T g

F

!

< ε.

Alternatively, T is of funny rank one if and only if there exists a sequence ( R n ) n ≥ 1

of funny towers with bases F n such that each set A ∈ B can be ε-approximated (as in (2)) by the union of some levels for each R n whenever n ≥ n ε . From this it easily follows that each system which is of funny rank one is AT. Indeed, the set of functions which are constant on levels of R n for some n ≥ 1 (and zero outside S R n ) are dense in L ¹ , and the set of those which are additionally positive is dense in L ¹ ₊ ; it follows that given f 1 , . . . , f l ∈ L ¹ ₊ (X, µ) and ε > 0 it is enough to take f = 1 1 F

(in (1)) for n ≥ 1 large enough.

The action T of G on (X, B , µ) induces a (continuous) unitary representation, called a Koopman representation, of G in the space L ² (X, B , µ) given by U T

f = f ◦ T g , f ∈ L ² (X, B , µ) and g ∈ G. We recall that a Koopman representation is said to have simple spectrum if for some f ∈ L ² (X, B , µ), L ² (X, B , µ) = span { f ◦ T g : g ∈ G } . Given f ∈ L ² (X, µ) we define its spectral measure σ f (or in a more precise notation, σ U

,f ) to be a finite Borel measure on the dual G b of G determined by

b

σ U

,f (g) = b σ f (g) :=

Z

G b

χ(g) dσ f (χ) = Z

X

f ◦ T g · f dµ, for all g ∈ G.

See [19], [20] for more information on the spectral theory of G-actions.

2. A criterion for a system to be non-AT

Let G be an Abelian discrete countable group which we assume to act on a probability standard Borel space (X, B , µ) as measure-preserving maps: g 7→ T g . Assume that P = { P 0 , P 1 } a partition of X (with P 0 ∈ B ). Through its P -names every point x ∈ X can be now coded: x 7→ π(x) = (x g ) g ∈ G where

x g =

0 if T g (x) ∈ P 0

1 if not.

Let Λ be a finite subset of G. By a funny word on the alphabet { 0, 1 } based on

Λ we mean a sequence (W g ) g ∈ Λ with W g ∈ { 0, 1 } , g ∈ Λ. For any two funny words

4 E. H. EL ABDALAOUI AND M. LEMA ´ NCZYK

W, W ^′ based on the same set Λ ⊂ G their Hamming distance is given by d Λ (W, W ^′ ) = 1

| Λ |

g ∈ Λ : W g 6 = W _g ^′ .

As noticed in [1] we then have the following extension of Dooley-Quas’ necessary condition for a system T = (T g ) g ∈ G to have the AT property [9].

Proposition 2.1. ([9]) Let (X, B , µ, T ) be an AT dynamical system. Then for any ε > 0 there exist a finite set Λ ⊂ G and a funny word W based on Λ such that

| Λ | µ { x ∈ X : d Λ (π(x) | Λ , W ) < ε }

> 1 − ε.

The contraposition in Proposition 2.1 yields automatically a criterion for a sys- tem to be non-AT. Some further work has been done in [1] to formulate a condition stronger than the negation of the assertion in Proposition 2.1 and which may be applied to many systems (the criterion obtained this way is an elaborated version of the method already used in [9]). We now present this criterion (Proposition 2.2 below) in its generality needed for this note.

A probability Borel measure ρ defined on G b is called a strong Blum-Hanson measure (SBH measure shortly) if the following holds

lim sup

k → + ∞

sup

Θ ⊂ G, | Θ | =k,(η

)

∈{ 1, − 1 }

Z

G b

√ 1 k

X

θ ∈ Θ

η θ · χ(θ)

2

dρ(χ) < 2.

Clearly the Haar measure m _{G b} of G b is an SBH measure; more generally, every absolutely continuous probability measure with density d ∈ L ¹ ( G, m b _{G b} ) satisfying k d k ∞ < 2, is an SBH measure. It is shown in [1] that each SBH measure is a Rajchman measure (i.e. its Fourier transform vanishes at infinity).

Proposition 2.2 ([1]) . Assume that a dynamical system (X, B , µ, T ) is ergodic and that there exists a partition P = { P 0 , P 1 } with the following properties:

i) There exists S ∈ Aut(X, B , µ) commuting with all elements T g , g ∈ G, such that SP 0 = P 1 .

ii) The spectral measure σ _{1 1}

_{− 1 1}

is an SBH measure.

Then the system T = (T g ) g ∈ G is not AT.

3. Countable Abelian discrete group action with simple spectrum and without the AT property

In this section we shall present the proof of Theorem 1.1.

3.1. Cocycles for discrete spectrum actions of countable discrete Abelian groups. For completness, we now briefly present an extension of the classical spec- tral theory of compact group extensions of Z-actions to group actions.

Let G be a countable Abelian discrete group acting on a standard probability Borel space (X, B , µ) as measure-preserving maps: g 7→ T g ∈ Aut(X, B , µ). Assume that Γ is a compact metric Abelian (written multiplicatively) group with Haar measure m Γ . A cocycle associated to T = (T g ) g ∈ G with values in Γ is a measurable function ϕ : X × G → Γ which satisfies

(3) ϕ(x, g + g ^′ ) = ϕ(x, g)ϕ(T g x, g ^′ )

for a.e. x ∈ X and all g, g ^′ ∈ G. We will also write ϕ g (x) instead of ϕ(x, g). Then the Γ-extension of T associated to ϕ is the G-action T ϕ = ((T ϕ ) g ) _{g ∈ G} defined on (X × Γ, µ ⊗ m Γ ) by

(T ϕ ) g : X × Γ → X × Γ, (x, γ) 7→ (T g x, ϕ g (x)γ).

The space L ² (X × Γ, µ × m Γ ) can be decomposed as

(4) L ² (X × Γ, µ × m Γ ) = M

χ ∈ b Γ

L χ , where each of the subspaces L χ := { f N

χ : f ∈ L ² (X, µ) } is (U T

) _g -invariant, and the restriction of the Koopman representation U T

of G to L χ is, via the map

(5) f ⊗ χ 7→ f, f ∈ L ² (X, µ),

unitarily equivalent to the G-representation V ϕ,T,χ = (V ϕ,T,χ,g ) _{g ∈ G} given by V ϕ,T,χ,g : L ² (X, µ) → L ² (X, µ), V ϕ,T,χ,g (f )(x) = χ(ϕ g (x))f (T g x), x ∈ X.

It follows that the spectral properties of T ϕ are determined by the spectral proper- ties of V ϕ,T,χ , χ ∈ Γ. b

Assume now that Γ = Z 2 = { 1, − 1 } . Set

S : X × Z 2 → X × Z 2 , S(x, ε) = (x, − ε)

and notice that S commutes with all automorphisms (T ϕ ) g , g ∈ G, i.e. S is in the centralizer of T ϕ . In this case the decomposition (4) is given by L ² (X × Z 2 , µ ⊗ m _Z

) = L 0 ⊕ L 1 where L 0 = { F ∈ L ² (X × Z 2 , µ ⊗ m _Z

) : F ◦ S = F } and L 1 = { F ∈ L ² (X × Γ, µ ⊗ m _Z

) : F ◦ S = − F } . Thus the restriction of the Koopman representation U T

on L 1 is unitarily equivalent to the representation V = (V g ) g ∈ G on L ² (X, µ) given by

(V g (f ))(x) = ϕ g (x)f (T g x), for all x ∈ X, g ∈ G.

Let P = { P 0 , P 1 } be the partition of X ×Z 2 given by P 0 = X ×{ 1 } , P 1 = X ×{− 1 } . Then 1 1 P

− 1 1 P

∈ L 1 , in fact

1 1 P

− 1 1 P

= 1 1 X ⊗ 1 1 _{ 1 } − 1 1 _{− 1 }

= 1 1 X ⊗ χ

where χ is the only non-trivial character of Z 2 ; in particular k 1 1 P

− 1 1 P

k L

= 1, so its spectral measure is a probability measure. In view of (5) we have

b

σ U

, 1 ¹

− 1 ¹

(g) = b σ V

, 1 ¹

(g) = h V ϕ,T,χ,g 1 1 X , 1 1 X i L

(X,µ) =

Z

X

χ(ϕ g (x)) dµ(x) = Z

X

ϕ g (x) dµ(x).

Moreover, SP 0 = P 1 . In this way we have proved the following.

Lemma 3.1. For any ergodic G-action T = (T g ) g ∈ G and its Z 2 -extension T ϕ for the partition P = { P 0 , P 1 } defined above we have:

(i) The element S defined above is in Aut(X × Z 2 , µ ⊗ m _Z

) and belongs to the centralizer of U T

.

(ii) The spectral measure σ ₁ 1

− 1 ¹

= σ U

, 1 ¹

− 1 ¹

satisfies (6) σ b _{1 1}

_{− 1 1}

(g) =

Z

X

ϕ g (x) dµ(x) for all g ∈ G.

6 E. H. EL ABDALAOUI AND M. LEMA ´ NCZYK

Comparing the above lemma with Proposition 2.2 we can see that the only thing which is missing is the fact that in general the measure σ 0 = σ _U

_{, 1 1}

_{− 1 1}

is not SBH, and to achieve non-AT property we will have to control the Fourier transform of σ 0 to obtain absolute continuity and a relevant boundedness of the density.

Recall also that if in addition the spectrum of the G-action T is discrete then by an obvious extension of Helson’s analysis from [17] we obtain that the spectral type of V has the purity law: it is either discrete, or continuous and purely singular, or equivalent to Haar measure m _{G b} .

3.2. Morse cocycles and Guenais’ constructions. We now present Guenais’

construction from [14]. Assume that (p n ) n ≥ 0 is an increasing sequence of prime numbers and let G = ⊕ n ≥ 0 Z/p n Z. Consider the action of G on its dual X = Q + ∞

n=0 Z/p n Z by translations T g x = x + g, where the space X is equipped with Haar measure µ = m X (which is the product of counting measures m _Z /p

Z on coordinates). The resulting G-action has discrete spectrum. Moreover, the action T = (T g ) g ∈ G has funny rank one. Indeed, to see it set

F n = { x ∈ X : x 0 = · · · = x n − 1 = 0 } , G n =

n − 1

M

k=0

Z/p n Z, n ≥ 0.

The sequence ( R n ) of funny Rokhlin towers (see (2)) is given by (T g F n ) g ∈ G

where F 0 = X , G 0 = { 0 } . Note that the union of levels of each such Rokhlin tower fills up the whole space.

As before we consider Γ = Z 2 and by a Morse cocycle one means a cocycle which will be constant on all levels of the funny Rokhlin towers (T g F n ) g ∈ G

, n ≥ 1, in the sense made precise below.

Take an arbitrary sequence of (ε n (j)) 0 ≤ j ≤ p

, n ≥ 0) with values in Z 2 in which ε n (0) = 1. Then define

ϕ(x, g) = ϕ g (x) = Y ∞ n=0

ε n (x n )ε n (x n + g n ) for all (x, g) ∈ X × G.

(7)

By (7) it follows directly that the cocycle identity (3) is satisfied and moreover for x ∈ F n and g ∈ G n we have

φ g (x) = Y ∞ k=0

ε k (g k ) =

n Y − 1 k=0

ε k (g k ),

so φ g is constant on F n for g ∈ G n (φ g (x) = φ g (0)) and in fact since ϕ(T h x, g)ϕ(x, g) = ϕ(x, g + h),

we also have ϕ g is constant on each level T h F n , h ∈ G n , whenever g ∈ G n : φ g | T

F

= φ g+h (0)φ g (0). (In fact, as shown in [14], each Morse cocycle is defined by a Z 2 -valued sequence (ε n (j)) 0 ≤ j ≤ p

, n ≥ 0).) The following has been observed in [14]:

(8) For each Morse cocycle ϕ, the representation V on L 1 has simple spectrum.

Now by the purity law it follows that once the spectral measure of 1 1 X (for

V ) is continuous, the spectrum on L 1 is continuous and therefore the Koopman

representation U T

has simple spectrum whenever ϕ is a Morse cocycle.

Set ε n (k) = k

p n

for 0 < k < p n and ε n (0) = 1, where k

p n

is the Legendre symbol, that is, it is equal to 1 if k 6 = 0 is a square modulo p n , − 1 if not and 0

p n

= 0.

Theorem 3.2 ([14]). The spectral measure σ 0 = σ _{V, 1 1}

of the function 1 1 X is the product measure N _∞

n=0 | P n | ² m _Z /p

Z where P n is defined on Z/p n Z by P n (x) = 1

√ p n

1 +

p X

− 1 k=1

k p n

e ^{− 2iπ}

! .

Moreover, σ 0 is equivalent to the Haar measure of the dual group G b with (9)

O ∞ n=0

| P n | ² m _Z /p

Z

!

b (g) = σ b 0 (g) for each g ∈ G.

It follows that the sequence N N

n=0 | P n | ² m _Z /p

Z

N ≥ 1 of measures converges weakly to σ 0 and since σ 0 ≪ m X we obtain the following.

Corollary 3.3. The sequence N N

n=0 | P n | ²

N ≥ 1 converges weakly in L ¹ (X, m X ) to _dm ^dσ

.

Notice that the sequence ( | P n | ² ) n ≥ 0 meant as polynomials on X (P n (x) = P n (x n ) for x ∈ X) is ultra flat, that is

k P n |k L

k P n k ^L

→ 1;

indeed, P n (0) = √ ¹ p

and for x 6 = 0 by the Gauss formula [3], [4]

P n (x) = 1

√ p n

1 +

p X

− 1 k=1

k p n

e ^{− 2iπ}

!

=

√ 1 p n 1 +

p X

− 1 k=0

e ^{− 2iπk}

^x/p

−

p X

− 1 k=0

e ^{− 2iπkx/p}

!

= 1

√ p n

1 + δ p

√ p n

x p n

with δ p

= 1 if p n ≡ 1 mod 4 or δ p

= i if p n ≡ 3 mod 4; whence

(10) 1 − 1

√ p n ≤ | P n (x) | ≤ 1 + 1

√ p n

. Recall also the following elementary fact.

Lemma 3.4. Let a ∈ (0, 1). Then we have Y ∞

n=1

(1 + a ⁿ ) ≤ exp a

1 − a

.

In view of (10) and Corollary 3.3

(11) dσ 0

dm X ≤ Y ∞ n=0

1 + 1

√ p n

2

.

8 E. H. EL ABDALAOUI AND M. LEMA ´ NCZYK

Let us choose now the sequence (p n ) n ≥ 0 so that √ p n ≥ 5 ⁿ⁺¹ for any n ≥ 0. By Lemma 3.4 we have

Y ∞ n=0

(1 + 1

√ p n

) ≤ e ^1/4 . Hence by (11), _dm ^dσ

X

≤ e ^1/2 < 2 and the proof of Theorem 1.1 is complete.

Remark 3.5. The action of the group G = L _∞

n=0 Z/p n Z on its dual by translations (as above) is a particular case of so called (C, F )-actions of the group G (see [7]), all such actions have the funny rank one property. It would be interesting to see whether it is possible to carry out Guenais’s construction of “good” Morse cocycles over a weakly mixing (C, F )-action of the group G to obtain a system which is not AT; as Morse cocycles over (C, F )-actions yield systems which have simple spectrum on L 1 we have additionally to know that the spectral types on L 0 and L 1 are mutually singular. Such a construction would be an analog of Ageev’s constructions [2] for Z-actions. It would be even more interesting if we could carry out the above over mixing (C, F )-actions of the group G constructed by Danilenko in [6].

Acknowledgment 1. The authors would like to thank J.-P. Thouvenot and F.

Parreau for fruitful discussions on the subject.

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Department of Mathematics, University of Rouen, LMRS, UMR 60 85, Avenue de l’Universit´ e, BP.12, 76801 Saint Etienne du Rouvray - France

E-mail address: [email protected]

Faculty of Mathematics and Computer Science, Nicolas Copernicus University, Chopin street 12/18, 87-100, Toru´ n, Poland

E-mail address: [email protected]